I Would imagine the rule will differ for different online card rooms as it does for land based card rooms. Some places may let you play all the chips you have left. Most places will not deal a player a hand unless they can call a full bet, I.e. the size of the big blind. I would think this would extend to the small blind. I do not think a rule would vary between full ring games and head up games. If it is a tournament you do have to get a all of a players chip so a player gets hands no matter his stack size.
Personally I used to play a lot of cash head up poker online. If a guy got low on chips I would stop playing and text them they need to reload if they want to keep playing. If your upside is some guys last few bucks and their upside is more like a hundred or two of yours, your not smart to play.
A good answer to your question is "since your playing head up, you have total control on rather or not you take a hand against a guy with only a small stack"
And specifically answering:"Does Player 2 still has to match big blind or does the game go to the Showdown?" The player is all in, therefore the game goes to showdown.